Question 312448
<font face="Garamond" size="+2">


The formula to calculate future value, *[tex \Large A], of an investment, *[tex \Large P], at *[tex \Large r] annual rate for *[tex \Large t] years, where *[tex \Large e] is the base of the natural logarithms and *[tex \Large r] is expressed as a decimal fraction is:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ A\ =\ Pe^{rt}]


Actually, you don't care what the initial investment is for this problem.  All you care about is that the ratio *[tex \Large \frac{A}{P}\ =\ 3], which is to say:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ e^{rt}\ =\ 3]


Take the natural log of both sides:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ln(e^{rt})\ =\ \ln(3)]


Use


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \log_b(x^n)\ =\ n\log_b(x)]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ rt\ln(e)\ =\ \ln(3)]


Use


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ln(e)\ =\ 1]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ rt\ =\ \ln(3)]


but for your case *[tex \Large r\ =\ 0.04], hence


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ t\ =\ \frac{\ln(3)}{0.04}]


A (very) little calculator work and you have your approximate answer.


Compare your result here to the decision whether to take a Lotto Jackpot payment in 26 equal installments or to take half of the money in one lump sum.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
</font>